Question

The motion of planets in the solar system is an example of the conservation of

• A. mass
• B. linear momentum
• C. energy
• D. angular momentum

Answer 1

Answer: (D)

From Kepler’s second law a line joining any planet to the sun sweeps out equal areas in equal times, that is, the areal speed of the planet remains constant.



$dA=$ area of curved triangle $SAB$

$=\frac{1}{2}(AB\times SA)$

$=\frac{1}{2}(rd\theta \times r)=\frac{1}{2}{r}^{2}d\theta$

The instantaneous areal speed of the planet is

$\frac{dA}{dt}=\frac{1}{2}{r}^2\frac{d\theta }{dt}=\frac{1}{2}{r}^2\omega$

where $\omega$ is an angular speed. Let $J$ be angular momentum of planet about sun

$J=I\omega =m{r}^2\omega$

$\frac{dA}{dt}=\frac{J}{2m}$

From Kepler’s law areal speed is constant, therefore angular momentum $J$ is constant.

Hence, Kepler’s second law is equivalent to conservation of angular momentum.